We also hope that our strategy can be extended to different settings. Let us mention the cases where \(\operatorname{tr}\left(A(x) D^2 u\right)\)  is replaced by a nonlocal diffusion operator, or \(Du, D^2 u\)  become the horizontal gradient and horizontal Hessian respectively induced by a family of Hörmander vector fields, as in [1]}. We stress again that, to achieve full \(L^q\) -maximal regularity, the adaption of the methods in [2]} are by no means obvious. On the other hand, the strategy presented here basically relies on Liouville theorems, which can be obtained in rather different ways. We also mention that, for parabolic problems, an exhaustive picture on the maximal regularity of solutions is still missing, despite some partial results appeared in [3]}.
